416 Mr. G. Boole o« the Solution of a 



and substituting 



= ^^^)"'"(^) {?(0}'"^'-"(;§)'Vw-^"'-^'-"^->2a^N (6.) 



Cq, c„ &c. being arbitrary constants, and the summation de- 

 noted by 2 extending from A=0 to X = r. And a little atten- 

 tion to the above result will show that it may be written thus, 



«={f(0}-"{|)''^, 

 when w is the complete integral of the differential equation 



(p{t)-T7'(io—{m + r — 7i)f'{t)w = Xcxi^. 



at^ 

 In this equation let us replace <p{t) by its value — +l>t + c, 



and integrate the result in a series. We shall thus find 



lo^Ao + A^t + A/^ + A^/^ + kc, . . . (7.) 

 Aq being a new arbitrary constant, and the remaining coeffi- 

 cients determined by a law whose expression is 



Cp-i—0{p—m — r + 7t—l)Aj,-i—ai^- — m — ri-?i — l ) A^_2 

 A„= ^ '— W 



Now since the values of c^,_i range from c^ to Cr, i. e. from 

 /j=l tojt; = r+l, after which they vanish, it follows from the 

 above that the values of A^ will be arbitrary up to A,.+i, after 

 which they will be formed from the preceding coefficients by 

 a law whose expression is 



b{p—m — r-\-n—\)Ap^i + a(^—m—r-irn — \ )Ap-2 



p,= ^ £ (9.) 



P cp 



Now /«/^ , /, ^ \-"'( d\ 



/ d \" 

 When, moreover, the operation ( -77 ) 'S performed upon the 



successive terms of to, the first r terms will vanish, and we 

 shall have 



/a^2 \r,r(r+i) . r()'+2)n 



+^-^-r(3)— "J } 



the two first coefficients of the bracketed series being arbitrary 



